Mathematical classification of algebraic structures using random walks theory
The 'Random walks on hyperbolic groups' (RWHG) project, funded by the EU, explored the area where three different fields of mathematics converged: probability theory, algebra and geometry. Although these fields may seem quite isolated, random walk theory has a major role to play in the area of their convergence. It is a branch of probability theory that mathematically represents a path comprising a succession of random steps. In general, there are two points of view with regard to the relationship between probability theory and algebra and geometry. The probabilistic viewpoint addresses questions on the impact of the underlying structure on the behaviour of the corresponding random walk. Random walks, however, describes the dynamics or behaviours of the structure-of-interest. In particular, algebraic and geometric properties can be classified based on the behaviour of their corresponding random walks. A specific aim of the RWHG project was to find a classification of algebraic structures that is defined by an analogue of the central limit theorems. The project set out to give a detailed and exhaustive list of all the abstract objects that satisfy their definition. The RWHG project thus realised the first step towards this ambitious objective. A central limit theorem was successfully proved for co-compact fuchsian hyperbolic groups. The proof proposed by the project could be applied to large classes of mathematical objects or groups. Unfortunately, the project ended prior to maturity, but it is hoped that further work will be done in this very complex field of mathematics.
